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Using the $f\sigma_8(z)$ redshift space distortion (RSD) data, the$\sigma_8^0-\Omega_m^0$ tension is studied utilizing a parameterization ofgrowth rate $f(z) = \Omega_m(z)^\gamma$. Here, $f(z)$ is derived from theexpansion history $H(z)$ which is reconstructed from the observational Hubbledata applying the Gaussian Process method. It is found that different priors of$H_0$ have great influences on the evolution curve of $H(z)$ and the constraintof $\sigma_8^0-\Omega_m^0$. When using a larger $H_0$ prior, the low redshifts$H(z)$ deviate significantly from that of the $\Lambda$CDM model, whichindicates that a dark energy model different from the cosmological constant canhelp to relax the $H_0$ tension problem. The tension between our best-fitvalues of $\sigma_8^0-\Omega_m^0$ and that of the \textit{Planck} 2018$\Lambda$CDM (PLA) will disappear (less than $1\sigma$) when taking a prior for$H_0$ obtained from PLA. Moreover, the tension exceeds $2\sigma$ level whenapplying the prior $H_0 = 73.52 \pm 1.62$ km/s/Mpc resulted from the HubbleSpace Telescope photometry. By comparing the $S_8 -\Omega_m^0$ planes of ourmethod with the results from KV450+DES-Y1, we find that using our method andapplying the RSD data may be helpful to break the parameter degeneracies.

Testing the effect of $H_0$ on $f\sigma_8$ tension using a Gaussian Process method